Finite Group Databasebeta

← #72220235

Subgroups — #72220235

Order 2592.

318 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13049411C212yesmaximal
2#13049399C211yesmaximal, characteristic
3C2×#66472—22no—
4#10620150C2211yescharacteristic
5C3⋊(C3⋊#622)C411yescharacteristic, complemented
6C32×#622C411yescharacteristic, complemented
7#88924—62no—
8C2×C3⋊(C3⋊(C3⋊D3))D411yescharacteristic
9#66472—42no—
10C32×C3⋊Dic3C2×C411yescharacteristic
11C32×C3⋊Dic3Q811yescharacteristic
12#63769—91nomaximal
13#63746—91nomaximal
14#6289—62no—
15#6289—121no—
16C3⋊#622—62no—
17C3×#622—22no—
18C2×C34C4⋊C411yescharacteristic
19C3⋊(C3⋊(C3⋊D3))D811yescharacteristic
20C3⋊(C3⋊(C3⋊D3))Q1611yescharacteristic
21C4⋊#157—92no—
22C4⋊#157—92no—
23#3787—91no—
24C8×C3⋊D3—91no—
25C2×C3⋊(C3⋊D3)—62no—
26C2×C3⋊(C3⋊D3)—121no—
27C2×C3⋊(C3⋊D3)—244no—
28C3×C3⋊Dic3—22no—
29C3×C3⋊Dic3—41no—
30C32×Dic3—62no—
31C32×Dic3—121no—
32C34#10711yescharacteristic, complemented, Sylow
33C2×#157—182no—
34C2×#157—182no—
35C2×#157—368no—
36C4×C3⋊D3—91no—
37C4×C3⋊D3—91no—
38#624—364no—
39#622—91no—
40#622#15711yescharacteristic, complemented
41C32×C8—91no—
42C3⋊(C3⋊C8)—91no—
43C2×C33—22no—
44C2×C33—41no—
45C2×C33—84no—
46C3⋊(C3⋊D3)—62no—
47C3⋊(C3⋊D3)—62no—
48C3⋊(C3⋊D3)—121no—
49C3⋊(C3⋊D3)—121no—
50C3⋊(C3⋊D3)—244no—
51C3⋊(C3⋊D3)—244no—
52D3×C8—542no—
53C2×C3⋊D3—91no—
54C2×C3⋊D3—91no—
55C2×C3⋊D3—364no—
56C2×C3⋊D3—368no—
57C2×C3⋊D3—724no—
58C2×C3⋊D3—724no—
59C2×C3⋊D3—722no—
60#157—362no—
61#157—362no—
62#157—3616no—
63C32×C4—91no—
64C32×C4—91no—
65C3⋊Dic3PSU(3,𝔽2)11yescharacteristic
66C3⋊Dic3C2×#15711yescharacteristic
67C3×Dic3—124no—
68C3×Dic3—124no—
69#107—811noSylow
70C33—22no—
71C33—41no—
72C33—84no—
73C4×D3—542no—
74C4×D3—1081no—
75C3×C8—182no—
76C3⋊C8—542no—
77C2×C32—44no—
78C2×C32—48no—
79C2×C32—84no—
80C2×C32—84no—
81C2×C32—82no—
82C2×C32#377711yescharacteristic
83C2×C32C4⋊#15711yescharacteristic
84C3⋊D3—91no—
85C3⋊D3—91no—
86C3⋊D3—91no—
87C3⋊D3—91no—
88C3⋊D3—364no—
89C3⋊D3—364no—
90C3⋊D3—368no—
91C3⋊D3—368no—
92C3⋊D3—724no—
93C3⋊D3—724no—
94C3⋊D3—724no—
95C3⋊D3—724no—
96C3⋊D3—722no—
97C3⋊D3—722no—
98C2×C8—811no—
99C4⋊C4—812no—
100C2×D3—542no—
101C2×D3—1081no—
102C2×D3—2164no—
103C3×C4—182no—
104C3×C4—361no—
105Dic3—62no—
106Dic3—121no—
107C32—44no—
108C32—48no—
109C32—84no—
110C32—84no—
111C32—82no—
112C32#6374611yescharacteristic, complemented
113C32#6376911yescharacteristic, complemented
114C2×C4—811no—
115C2×C4—1622no—
116C8—91no—
117C8—811no—
118C2×C3—22no—
119C2×C3—41no—
120C2×C3—84no—
121D3—542no—
122D3—542no—
123D3—1081no—
124D3—1081no—
125D3—2164no—
126D3—2164no—
127C22—811no—
128C4—91no—
129C4—91no—
130C4—3242no—
131C3—22no—
132C3—41no—
133C3—84no—
134C2—811no—
135C2—811no—
136C2#1304978311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2#13049783
commutatorC32×C3⋊Dic3C2×C4
Fitting, socleC2×C34C4⋊C4
prime radicalC2×C3⋊(C3⋊(C3⋊D3))D4
cosocle#10620150C22
hypocenterC34#107
LCSG ▷ C32×C3⋊Dic3 ▷ C2×C34 ▷ C34
UCSC1 ◁ C2

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. C3⋊(C3⋊#622)C4semidirect
6. C32×#622C4semidirect
32. C34#107semidirect
40. #622#157semidirect
112. C32#63746semidirect
113. C32#63769semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#130494112
#130493992
#637699
#637469